big picture - Why is it useful to classify the vector bundles of a space?

I am not sure why you think the classification of vector bundles is a difficult problem. It quickly reduces to homotopy theoretic issues that are quite well understood (as well as anything in homotopy theory can be understood).

Vector bundles are important because they give natural invariants of manifolds (tangent bundle and its characteristic classes), and embeddings (normal bundle), as well as a natural place where various structures attached to a manifold live (e.g. metrics, connections, tensors).

Whether the classification of bundles matters depends on what you do. To make an analogy if you are studying PDE on $mathbb R^n$, which is a huge subject, then you might not care about classifications of manifolds at all; except that PDE do come in handy in classifying 3-manifolds. As such classification of vector bundles isn't used in most of the mathematics, rather it represents a basic result in topology.

In fact I am curious how often do you use classification of vector bundles (be that identification of $mathbb R^k$-bundles over $X$ with $[X, BO(k)]$ or the fact that complex line bundles are classified by first Chern class, or classification of bundles over complexes of dimension $le 4$ in terms of characteristic classes)? In my own research I have used all this extensively but my impression that this is quite rare.